JournalsrmiVol. 36, No. 7pp. 2209–2216

Two weight inequalities for positive operators: doubling cubes

  • Wei Chen

    Yangzhou University, China and Georgia Institute of Technology, Atlanta, USA
  • Michael T. Lacey

    Georgia Institute of Technology, Atlanta, USA
Two weight inequalities for positive operators: doubling cubes cover
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Abstract

For the maximal operator MM on Rd\mathbb{R}^{d}, and 1<p,ρ<1 < p,\rho < \infty, there is a finite constant D=Dp,ρD=D _{p, \rho } so that this holds. For all weights w,σw,\sigma on Rd\mathbb{R}^{d}, the operator M(σ)M(\sigma \cdot) is bounded from Lp(σ)Lp(w)L^{p}(\sigma )\to L^{p}(w) if and only if the pair of weights (w,σ)(w,\sigma) satisfy the two weight ApA_{p} condition, and this testing inequality holds:

QM(σ1Q)pdwσ(Q),\int_{Q} M(\sigma \mathbf 1_{Q})^{p} dw \lesssim \sigma(Q),

for all cubes QQ for which there is a cube PQP\supset Q satisfying σ(P)Dσ(Q)\sigma(P) \leq D\sigma(Q), and (P)ρ(Q)\ell(P) \geq \rho \ell(Q). This was recently proved by Kangwei Li and Eric Sawyer. We give a short proof, which is easily seen to hold for several closely related operators.

Cite this article

Wei Chen, Michael T. Lacey, Two weight inequalities for positive operators: doubling cubes. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2209–2216

DOI 10.4171/RMI/1197